| L(s) = 1 | + (1.41 + 0.0358i)2-s + (1.54 − 1.12i)3-s + (1.99 + 0.101i)4-s + (−0.00959 + 2.23i)5-s + (2.22 − 1.53i)6-s + (−1.00 − 1.00i)7-s + (2.82 + 0.215i)8-s + (0.203 − 0.627i)9-s + (−0.0937 + 3.16i)10-s + (1.22 + 0.625i)11-s + (3.20 − 2.08i)12-s + (−4.81 − 1.56i)13-s + (−1.38 − 1.45i)14-s + (2.49 + 3.47i)15-s + (3.97 + 0.405i)16-s + (−0.345 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0253i)2-s + (0.893 − 0.649i)3-s + (0.998 + 0.0507i)4-s + (−0.00429 + 0.999i)5-s + (0.909 − 0.626i)6-s + (−0.380 − 0.380i)7-s + (0.997 + 0.0760i)8-s + (0.0679 − 0.209i)9-s + (−0.0296 + 0.999i)10-s + (0.370 + 0.188i)11-s + (0.925 − 0.603i)12-s + (−1.33 − 0.433i)13-s + (−0.370 − 0.389i)14-s + (0.645 + 0.896i)15-s + (0.994 + 0.101i)16-s + (−0.0839 − 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.04905 - 0.192004i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.04905 - 0.192004i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0358i)T \) |
| 5 | \( 1 + (0.00959 - 2.23i)T \) |
| good | 3 | \( 1 + (-1.54 + 1.12i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (1.00 + 1.00i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.22 - 0.625i)T + (6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (4.81 + 1.56i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.345 + 2.18i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.843 + 0.133i)T + (18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (3.78 + 1.93i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (2.89 + 0.458i)T + (27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (-2.14 + 2.94i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (7.25 + 2.35i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-5.88 - 1.91i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.93iT - 43T^{2} \) |
| 47 | \( 1 + (1.11 - 7.00i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.00879 + 0.00638i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.96 - 1.00i)T + (34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (-11.7 - 5.98i)T + (35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (1.81 - 2.49i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-8.71 + 6.32i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.00 - 9.82i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (13.5 - 9.82i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 8.09i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.686 + 2.11i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (5.39 + 0.854i)T + (92.2 + 29.9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.46182157064571454974996489830, −10.43136389583885471876497844106, −9.626911722882685055153553228588, −8.041763206803610621306710905225, −7.28180658335386013448891232068, −6.77847298249984099615796771125, −5.49459250277309093427318882221, −4.06692798633509340528753021138, −2.93330728648614204863089067557, −2.19661801593736009015334916371,
2.05652740759128110002676920400, 3.37295930938782173164379976202, 4.27841451781150989511110692573, 5.21642071528014717073150065226, 6.31745308887028710327374758407, 7.60381485180195349953903490188, 8.649458150605739001924413730310, 9.488823135746854973607755985697, 10.21216377340066904006010324527, 11.69260529562620049030085557006
